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Bisexual branching processes to model extinction
conditions for Y-linked genes
Miguel González, Rodrigo Martı́nez, Manuel Mota
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Miguel González, Rodrigo Martı́nez, Manuel Mota. Bisexual branching processes to model
extinction conditions for Y-linked genes. Journal of Theoretical Biology, Elsevier, 2009, 258
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Author’s Accepted Manuscript
Bisexual branching processes to model extinction
conditions for Y-linked genes
Miguel González, Rodrigo Martínez, Manuel Mota
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doi:10.1016/j.jtbi.2008.10.034
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Cite this article as: Miguel González, Rodrigo Martínez and Manuel Mota, Bisexual
branching processes to model extinction conditions for Y-linked genes, Journal of Theoretical Biology (2008), doi:10.1016/j.jtbi.2008.10.034
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Bisexual branching processes to model
extinction conditions for Y-linked genes Miguel González Rodrigo Martı́nez Manuel Mota
Department of Mathematics, University of Extremadura, Badajoz 06071, Spain.
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Abstract
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In a two-sex monogamic population, the evolution of the number of carriers of
the two alleles of a Y-linked gene is considered. To this end, a multitype bisexual
branching model is presented in which it is assumed that the gene has no influence
on the mating process. It is deduced from this model that the average numbers
of female and male descendants per mating unit constitute the key to determining
the extinction or survival of each allele. Moreover, the destiny of each allele in the
population is found not to depend on the behaviour of the other.
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Key words: Sex-linked inheritance, Y chromosome, two-dimensional bisexual
stochastic model, perfect fidelity mating, survival.
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Introduction
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Sex determination in most mammal populations, including humans, is due to a
pair of chromosomes denominated X and Y. Females have XX chromosomes,
while males have two distinct chromosomes, XY. The SRY gene on the Y
chromosome determines the development of the testes, and hence maleness (see
Wallis et al. (2008)). The Y chromosome has very few genes in comparison with
the X chromosome, but recent investigations have shown the importance of
some Y-linked genes in populations of humans (see, for example, the web page
www.nature.com/nature/focus/ychromosome/, Quintana-Murci and Fellous
(2001), or Hughes et al. (2005)) and other animals which have the XX/XY
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The research was supported by the Ministerio de Educación y Ciencia and the
FEDER through the Plan Nacional de Investigación Cientı́fica, Desarrollo e Innovación Tecnológica, grant MTM2006-08891.
Corresponding author. e-mail: [email protected]
Preprint submitted to Elsevier
27 October 2008
type of sex determination system (see, for example, Bernardo et al. (2001),
Gutiérrez and Teem (2006), or the review by Charlesworth et al. (2005)).
Moreover this chromosome has the particularity of being male-specific and
haploid, and of having a region (the non-recombining region, NRY, being 95%
of the chromosome in humans – see, for example, Krausz et al. (2004) or
Graves (2006)) which escapes recombination. These unique properties of the
Y chromosome have important consequences for its population genetics: the
NRY region passes down from father to son largely unchanged, and is therefore
very useful for studying how populations have evolved. By examining the
differences between modern Y chromosomes (such as DNA polymorphisms),
one can attempt to reconstruct a history of paternal lineages. There have been
many studies in this sense in the context of populations of humans (e.g., Hurles
et al. (1998), Quintana-Murci et al. (2001), Hurles et al. (2002), or Rosa et al.
(2007)) and other species (e.g., Tosi et al. (2002), Hellborg et al. (2005), or
Geraldes et al. (2005)).
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Another singular question associated with the Y chromosome is that of the
microdeletions of this chromosome’s long arm (Yq). These Yq deletions define
three regions collectively known as AZF (azoospermia factors), with deletions
of the AZFc region being the commonest. The Yq deletion is associated with
males with fertility problems (for a review, see Krausz et al. (2003)), but many
cases have been reported in which the natural transmission of this genetic
defect from fathers to sons has occurred (see e.g., Saut et al. (2002), Calogero
et al. (2002) or Kuhnert et al. (2004)). Related to this question, Krausz and
McElreavey (2001) state: “The pathogenetic significance of Y chromosome
microdeletions is spermatogenic failure and not infertility. AZF deletions can
be associated with oligozoospermia which usually leads to infertility but it
can also be associated with normal couple fertility. For the interpretation of
any new observations about the transmission of Yq deletions we should keep
in mind that fathering one or more children is the expression of the fertility
status of both members of the couple”. Obviously, determining the evolution
of the number of males with this genetic defect in a human population is an
important medical problem (see, for example, Patsalis et al. (2002) or Fitch
et al. (2005)), but it has also been investigated in other species (e.g., Toure
et al. (2004)).
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The surname is another characteristic which can be seen as Y-linked in humans. There have been some recent studies aimed at determining the relationship between surnames and Y-chromosome lineages (e.g., King et al. (2006)
or Bowden et al. (2008)).
Other organisms have a mirror image of the XX/XY sex determination system,
with males being homogametic and females heterogametic. To avoid confusion,
these sex chromosomes are deniminated Z and W. Thus in birds, snakes, and
butterflies, for example, females have ZW sex chromosomes, and males have ZZ
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sex chromosomes. Now, Y-linked problems become W-linked (see, for example,
Ogawa et al. (1998), Yamada et al. (2004), or Abe et al. (2008)). For simplicity
in notation, throughout this paper only the XX/XY sex determination system
will be referred to, although the development and the results are equally valid
for the ZZ/ZW system.
Appropriate mathematical models are needed to understand the evolution
of Y chromosome lineages (for example, to help solve the problem of Ychromosomal Adam), Yq deletions, or other Y-linked genes.
Branching processes naturally come to mind in this context. These are stochastic models which arise in the description of population dynamics, being of
particular use in describing the extinction/growth of populations (see Haccou
et al. (2005)). Branching models have been applied to many biological problems in such fields as epidemiology, genetics, and cell dynamics. Examples
include the evolution of infectious diseases (e.g., Mode and Sleemam (2000),
Ball et al. (2004), or Garske and Rhodes (2008)), population genetics (e.g.,
Campbell (2003) or Iwasa et al. (2005)), and stem cells (e.g., Yakovlev and
Yanev (2006)). Further examples are reviewed in the recent monographs of
Kimmel and Axelrod (2002) and Pakes (2003), and in the communication of
Caron-Lormier et al. (2006).
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The simplest branching models, the Galton-Watson process and the Markov
branching process, have been used to model Y-chromosome lineages and their
female analogues – mitochondrial DNA lineages (see O’Conell (1995) and
Neves and Moreira (2006)). But more accurate models are needed in which all
the phases of sexual reproduction can be considered, including the interaction
between females and males in producing offspring. In González et al. (2006),
a model was presented that describes the evolution of the number of carriers
of the two alleles of a Y-linked gene in a two-sex monogamic population. In
that paper, it was considered that the characters controlled by such a gene can
have some influence on the mating process of the species, with females having
a preference for males carrying one of the alleles of the gene. It was shown
that this preference can sometimes be definitive in determining the survival
of the different genotypes in the population.
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However, females’ choice of mate is not always conditioned by the males’ genotype. Most Y-linked characters do not appear in the phenotype of the males,
or, even if they do, are not decisive at the time of mating (for example, in the
case of Yq deletions or many of the NRY region genes). In these situations
it seems more realistic to consider a model where females choose their mates
without caring about what their genotype is, i.e., each female makes a blind
choice of the genotype of her mate.
In this paper, we present a multitype bisexual branching process to model the
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evolution of the number of carriers of each allele of a Y-linked gene or of Y
chromosome lineages in a two-sex monogamic population, assuming that this
gene or characteristic does not influence the mating process. We focus our
attention on studying the possibility of fixation of one of the alleles of the
gene, and on determining the destiny of the gene in the population.
The reset of the paper consists of four sections. In Section 2 we provide the
definition of the Y-linked Bisexual Branching Process assuming that females
choose the genotypes of their mates blindly. We consider Y-linked genes with
two alleles. We also present some basic properties of the model. In Section
3 we study the extinction problem for both the whole population and each
genotype separately, in the latter case taking into account the behaviour of the
other genotype. For some critical situations, we simulate the evolution of the
population and conjecture its long term behaviour. In Section 4 we provide
some concluding remarks. Finally, Section 5 presents the proofs of the results.
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The model
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Consider a human or other animal diploid population with non-overlapping
generations, where the sex of an individual is determined by a pair of chromosomes X and Y. A female (F) will have XX chromosomes, while a male (M)
will have XY chromosomes.
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Consider also a gene with a pair of alleles, R and r, linked to the Y chromosome. The character defined by this gene is exclusive to males, which will be
designated by MR or Mr according to which allele they carry. We will suppose
however that they both have the same phenotype, so that mating is not influenced by the gene and females choose the genotype of their mates blindly.
Moreover, perfect fidelity (monogamy) is assumed at the time of mating, i.e.,
each individual mates with only one individual of the opposite sex, provided
that some of them are still available. Monogamy is the mating option in some
birds and mammals, and in particular in most human populations, and is
therefore well suited to the specific problems which motivated the present
work (e.g., Yq deletions, or the history of paternal lineages).
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A couple will be classified as FMR or FMr according to the genotype of the
male. Following the inheritance rules, FMR mating units can generate females
and MR males, while FMr mating units can produce females and Mr males.
We shall model the evolution of such a population by means of a two-type
bisexual branching process in discrete time. To define this model, assume that
it is known how many couples there are in the present generation, labeled
in the definition as generation n. The objective is to determine the number
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of females, males, and mating units of each genotype in the next generation,
labeled n + 1. To this end, two phases will be considered: reproduction and
mating.
In the reproduction phase, each couple of generation n is assumed to produce
offspring independently of the other couples according to a random variable.
The probability distribution of these variables will be the same for all the
couples with a given genotype, irrespective of the generation they belong to,
and will be termed the reproduction law of that genotype. In general, FMR
and FMr couples are assumed to have differences in their reproductive capacities, and consequently their respective reproduction laws are assumed to be
different. This is an appropriate assumption in such problems as Yq deletions
associated with male fertility problems. The important parameters will be the
average numbers of offspring produced by FMR and FMr couples, denoted
here by mR and mr , respectively.
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With probability α an offspring will be a female, and consequently with probability 1−α it will be a male. It is assumed that the genotype has no influence
on the sex determination, so that α is the same for both genotypes. Therefore, the average number of females and males per FMR couple will be αmR
and (1 − α)mR , respectively, and these values will be αmr and (1 − α)mr ,
respectively, for an FMr couple.
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The female offspring of all the couples in generation n gives the total number
of females in generation n + 1, which will be denoted by Fn+1 . On the basis of
the genetic rules described above, the male offspring of all the FMR couples
in generation n gives the total number of MR males in generation n + 1,
denoted by MRn+1 . Analogously, the male offspring of all the FMr couples in
generation n gives the total number of Mr males in generation n + 1, denoted
by Mrn+1 .
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In the mating phase, the number of couples of each genotype formed in generation n + 1 is calculated, taking into account that generations do not overlap
and provided that in the population there are Fn+1 females, MRn+1 males with
R genotype, and Mrn+1 males with r genotype. We will denote by ZRn+1 and
Zrn+1 the total number of FMR and FMr couples, respectively, in generation
n + 1.
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The point here is the perfect fidelity mating considered in the premises of the
model. According to this assumption, the total number of couples in generation
n + 1 is the minimum of the number of females, Fn+1 , and the total number
of males, MRn+1 + Mrn+1 .
If the number of females exceeds that of males, then all the males mate, and
consequently the number of couples of a genotype will be equal to the number
of males of that genotype, i.e., ZRn+1 = MRn+1 and Zrn+1 = Mrn+1 . On the
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contrary, if the number of females is less than or equal to the number of males,
all the females mate and must choose the genotype of their mates. However,
since all the males have the same phenotype, each of the Fn+1 females makes
a blind choice between the MRn+1 males with R genotype and the Mrn+1
males with r genotype. Thus, the number of F MR couples, ZRn+1 , will be
given by a hypergeometric distribution of parameters Fn+1 , MRn+1 + Mrn+1 ,
and MRn+1 , i.e. Fn+1 males are selected from all males of generation n + 1
where MRn+1 males have the R genotype. The rest of the couples will have r
genotype, i.e., Zrn+1 = Fn+1 − ZRn+1 .
Obviously, the symmetry in the mating process means that it would be equivalent to consider Zrn+1 given by a hypergeometric distribution of parameters
Fn+1 , MRn+1 + Mrn+1 , and Mrn+1 , and the number of remaining couples up
to Fn+1 would be ZRn+1 . This symmetry is the main difference between the
present model and that with preference in the mating introduced in González
et al. (2006). In that model, the genetic character controlled by the gene does
appear in the phenotype, and females prefer males carrying one of the alleles of the gene. The consequent lack of symmetry in the mating had a great
influence on the results.
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The bivariate sequence {(ZRn , Zrn )}n≥0 indicating the generation-by-generation
evolution of the number of mating units will be termed a Y-linked bisexual
branching process.
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From the definition of the model, the number of couples of each genotype in the
next generation depends only on the present number of mating units, and not
on the number of ancestors that belonged to past generations. Moreover, since
each reproduction law remains the same over the generations, the transitions
from one generation to another are homogeneous, i.e., they do not depend on
the generation. The process {(ZRn , Zrn )}n≥0 is therefore a homogeneous twotype Markov chain. However, the general theory of Markov chains, based on
transition probabilities, does not lead to any significant results for this model.
It is therefore necessary to attempt a specific methodological approach that is
typical of branching processes.
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The model introduced here shares some basic properties with the model with
mating preference presented in González et al. (2006). These are properties
which do not depend on the way the females choose their mates, and were
proved in that paper. The following are four of these properties which are
germane to the present work:
P 1 If in some generation there are no mating units of a particular genotype
then, from this generation on, the mating units and the males of that genotype no longer exist. This is known as extinction of this particular genotype
which implies the fixation of the allele determining the other genotype. The
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extinction of the whole population occurs when there are no mating units of
any genotypes.
P 2 Provided that there exist couples of both genotypes in the present, there
is a positive probability of reaching any number of mating units of each genotype in future generations. In particular, the fixation of one of the genotypes
or even the extinction of the whole population are possible.
P 3 Each genotype presents the dual behaviour typical of branching processes: either it becomes extinct or the number of couples of this genotype
eventually reaches arbitrarily large values. The latter event is known as explosion or indefinite growth of this particular genotype. Consequently the whole
population also presents this duality. Thus, the survival of each genotype or
of the whole population is equivalent to their indefinite growth as generations
go by, with the possibility that, in the long term, their size tends to be in
the neighbourhood of one or more positive values having to be discarded. Although this property might seem unrealistic, it merely expresses what would
be the ideal long term evolution of a population when its development is not
constrained by any external bound.
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P 4 If the fixation of a particular allele has occurred, the corresponding genotype evolves essentially as a Bisexual Branching Process (BBP) with perfect
fidelity mating and the reproduction law of the surviving genotype. Analogously, if both genotypes have the same reproduction law, the evolution of the
total number of couples can also be modeled by a BBP with perfect fidelity
mating.
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With respect to this last property, it has to be remarked that the present
model in some way inherits some of the properties of the BBP, a model that
has been extensively investigated in the literature on Branching Processes.
One may cite as examples work on the extinction probability (e.g., Daley
(1968), Hull (1982, 1984), Bruss (1984), Daley et al. (1986), or Alsmeyer and
Rösler (2002)), the long term behaviour (e.g., Bagley (1986) or González and
Molina (1996, 1997)), and inference problems (e.g., González-Fragoso (1995),
Molina et al. (1998), Alsmeyer and Rösler (1996), or González et al. (2001)).
Two interesting recent reviews are Hull (2003) and Haccou et al. (2005).
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Before concluding this section, it is worth comparing the Y-linked bisexual
branching process with the classical genetic model of Wright-Fisher with which
it shares some common features. In particular, both are Markov chains and
consider diploid populations with non-overlapping generations. However there
are also many differences. The most important is that, in the Wright-Fisher
setting, the population size remains constant over the generations, reflecting a
regulation of population size by external resources. The Wright-Fisher model
may therefore not be the best model for a natural population. The Y-linked
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bisexual model, however, does allow indefinite growth. With respect to the
mating, the Wright-Fisher model considers this to be random in the sense
that every allele in a generation could come from every possible couple in the
previous one owning that allele. In the Y-linked bisexual process, the number
of couples owning each of the alleles, established in the mating phase, is of
prime importance in determining the number of carriers of that allele in the
subsequent generation.
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The extinction problem
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In this section we deal with two problems of interest in both the genetics
and the population dynamics frameworks (e.g., Yq deletions, or the history of
paternal lineages). The first is to obtain a necessary and sufficient condition
for the whole population to become extinct almost surely. The other is to
analyze the destiny of the gene in the population. To this end, we study
some relevant events, such as the extinction of each genotype (mathematically,
{ZRn → 0} and {Zrn → 0}) and the survival or indefinite growth of each
genotype (mathematically, {ZRn → ∞} and {Zrn → ∞}). Survival is possible
in the presence or absence of the other genotype, so we also consider the
events {ZRn → ∞, Zrn → 0} (fixation of R genotype), {ZRn → 0, Zrn →
∞} (fixation of r genotype), and {ZRn → ∞, Zrn → ∞} (survival of both
genotypes), which are also of interest in themselves. We provide conditions
depending on the parameters mR , mr , and α.
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We will assume that at the time the population was first observed, i.e., in
generation 0, there were i couples with R genotype and j couples with r
genotype. In order to simplify the notation, we denote P (·|(ZR0, Zr0 ) = (i, j))
by P(i,j) (·). Even (i, j) will be dropped in this notation if there is no ambiguity.
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The extinction of the population
It is understood that the population becomes extinct if from some generation
on there are no mating units. For every (i, j) we denote the probability of
extinction of a population starting with i FMR and j FMr mating units by
Q(i,j) = P(i,j)(ZRn → 0, Zrn → 0).
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Let QR
i (resp. Qj ) be the probability of extinction of a BBP with perfect fidelity
mating, reproduction law of R (resp. r) genotype, and i (resp. j) starting
mating units. From property P4, it follows that, for all i, j ≥ 0,
r
Q(i,0) = QR
i and Q(0,j) = Qj ,
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and therefore the results on the extinction of a BBP with perfect fidelity
mating (see Daley (1968)) apply here.
When i, j > 0, the following result provides a satisfactory answer to the problem of extinction.
Result 1 Let i, j > 0. Then Q(i,j) = 1 if and only if min{αmr , (1−α)mr } ≤ 1
and min{αmR , (1 − α)mR } ≤ 1.
We conclude that if the average number of females or males produced by a
mating unit of each type is less than or equal to one then the process becomes extinct almost surely. On the other hand, if at least one of the values
min{αmR , (1−α)mR } or min{αmr , (1−α)mr } is greater than one then, applying the results of a BBP, the corresponding genotype has a positive probability
of survival even without any contribution of the other genotype, and therefore
the population may survive too.
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The destiny of the gene in the population
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After determining a necessary and sufficient condition for the process to become extinct with probability one, we focus our interest on the long term
growth of the number of carriers of each allele.
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However, it would seem logical for there to exist some interdependence between
the behaviour of the two genotypes, so that, in the analysis of each genotype,
we must distinguish between whether the other has also an indefinite growth
or has become extinct. In particular, we have to deal with the events
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{ZRn → ∞, Zrn → 0}, {ZRn → 0, Zrn → ∞}, and {ZRn → ∞, Zrn → ∞}
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which represent the possible destinies of the two alleles in the population.
First we investigate the extinction of one genotype and the survival of the
other. Because of the symmetry in the role of the two genotypes, the study of
the fixation of each of them, {ZRn → ∞, Zrn → 0} and {ZRn → 0, Zrn →
∞}, leads to analogous conclusions. In both events, the surviving genotype
behaves from some generation on as a BBP with perfect fidelity mating (see
property P4). Hence, the theory of Daley (1968) about a BBP applies, and it
is immediate to deduce the following result.
Result 2 Let i, j > 0, then
(i) P(i,j) (ZRn → ∞, Zrn → 0) > 0 if and only if min{αmR , (1 − α)mR } > 1.
(ii) P(i,j) (ZRn → 0, Zrn → ∞) > 0 if and only if min{αmr , (1 − α)mr } > 1.
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Intuitively, this result states that a necessary and sufficient condition for a
genotype to have a positive probability of fixation is that both the female and
the male descendants per couple of such a genotype are on average greater
than one.
From the “if” parts of the above result, it is immediate to deduce the following
corollary relative to the growth of the number of carriers of each allele.
Result 3 Let i, j > 0, then
(i) If min{αmR , (1 − α)mR } > 1, then P(i,j) (ZRn → ∞) > 0.
(ii) If min{αmr , (1 − α)mr } > 1, then P(i,j) (Zrn → ∞) > 0.
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With respect to the simultaneous survival of both genotypes, i.e., the event
{ZRn → ∞, Zrn → ∞}, one has the following result.
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Result 4 Let i, j > 0.
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(i) If (1−α)mR ≤ 1 or (1−α)mr ≤ 1, then P(i,j) (ZRn → ∞, Zrn → ∞) = 0.
(ii) If αmR < 1 or αmr < 1, then P(i,j) (ZRn → ∞, Zrn → ∞) = 0.
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In Result 4 (i) we state that if the average number of male descendants per couple of either genotype is less than or equal to one, then there is null probability
of the simultaneous survival of both genotypes. The key to this behaviour lies
in the fact that the number of mating units of a genotype is bounded above
by the number of males of that genotype.
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Also, the probability of simultaneous survival of both genotypes is zero when
the average number of female descendants per couple of either genotype is less
than one. This is true, as indicated in Result 4 (ii), even though it could be
expected that the males of this genotype would mate with females generated by
mating units of the other genotype. For example, if one takes α < 0.5, αmR <
1, (1 − α)mR > 1, and αmr > 1, even though the number of FMr mating units
can grow indefinitely and the average number of MR male descendants per
FMR mating unit is greater than one, the FMR mating units become extinct
almost surely because not enough females are produced for all the population.
This is an amazing fact, which in no way can be considered intuitive. To
illustrate this scenario, we simulated 12 generations with (ZR0 , Zr0 ) = (50, 1),
α = 0.3, and the reproduction laws following Poisson distributions with mR =
2 and mr = 4. In Figure 1, we show a path of this process in which one
observes the pattern described above.
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The cases αmR = 1 and αmr = 1 are not considered in Result 4 (ii). If α ≥ 0.5,
and αmR = 1 or αmr = 1, then (1 − α)mR ≤ 1 or (1 − α)mr ≤ 1, respectively,
and Result 4 (i) guarantees that P(i,j)(ZRn → ∞, Zrn → ∞) = 0. However,
if α < 0.5, we have not proved whether there is positive or null probability
10
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200
60
ZR
Zr
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0
20
40
couples
100
50
individuals
150
FR
MR
Fr
Mr
0
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12
0
1
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3
generations
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12
generations
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Fig. 1. Path of a process in which αmR < 1, (1 − α)mR > 1, and αmr > 1.
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of the simultaneous survival of both genotypes. In order to conjecture the
possible behaviour of the process in such situations, we performed a Monte
Carlo simulation of ten batches of 10 000 processes until generation 1000, with
reproduction laws following Poisson distributions with means mR = 2.5 and
mr = 2.52, and probability for an offspring to be female α = 0.4. Under these
conditions αmR = 1, (1 − α)mR = 1.5 > 1, and αmr = 1.008 > 1. In all the
simulated processes we took (ZR0 , Zr0 ) = (3, 3). Table 1 lists the results for
the number of processes in each batch where both genotypes have survived by
a given generation. One observes how the number of such processes decreases
to zero over the course of succeeding generations, with similar records for
the ten batches. Also, Figure 2 shows the proportion of processes among the
100000 simulated ones in which both genotypes have survived by a given
generation. Actually, this plot provides an estimate of the probability that
both genotypes survive until generation n, with n = 1, 2, . . . . Figure 3 shows
the path of one process where both genotypes have survived until generation
1000. One observes geometric growth of the number of FMr mating units over
the course of succeeding generations. The evolution of the number of FMR
mating units, however, does not seem to fit any clear pattern, showing many
fluctuations. However, according to property P3, it must converge either to 0
or ∞. Since this path does not seem to have an indefinite growth, we guess
that eventually this path will become extinct, although it will take a great
number of generations as can be observed in the figure. We could therefore
conjecture that, in the situations with αmR = 1 or αmr = 1 not covered by
Result 4, P(i,j) (ZRn → ∞, Zrn → ∞) = 0.
s
u
n
a
d
e
m
t
p
e
c
c
A
Applying property P3, it is immediate to deduce the following corollary of
Results 2 and 4, related to the extinction of each allele.
Result 5 Let i, j > 0, then
(i) If min{αmR , (1 − α)mR } < 1, then P(i,j) (ZRn → ∞) = 0.
11
batch
1
2
3
4
5
6
7
8
9
10
generation 20
233
242
221
223
219
245
199
223
229
239
generation 40
55
58
49
54
45
48
49
51
56
49
generation 60
27
29
24
25
18
22
29
30
27
27
generation 80
18
17
15
17
11
13
14
21
14
17
generation 100
13
12
11
15
7
12
11
15
12
13
generation 200
3
3
2
4
2
4
4
7
6
6
generation 300
3
2
1
1
2
4
4
3
4
5
generation 400
3
2
1
1
1
1
1
2
4
2
generation 500
3
2
0
0
1
1
1
2
i
r
c
4
2
t
p
generation 1000
3
1
0
0
0
0
1
1
3
1
Table 1
Records of the number of processes in each batch where both genotypes have survived over the course of succeeding generations.
s
u
0.06
d
e
0.04
0.02
proportion
0.08
0.10
n
a
0.00
t
p
0
e
c
50
100
150
m
200
250
300
generations
Fig. 2. Proportion over the course of succeeding generations of simulated processes
in which both genotypes have survived.
c
A
(ii) If min{αmr , (1 − α)mr } < 1, then P(i,j) (Zrn → ∞) = 0.
Remark 1 From Results 3 and 5, one can infer that, roughly speaking, the
extinction or survival of each genotype depends only on its own reproductive
abilities, without any influence of the other genotype.
Finally, our interest now is to find conditions guaranteeing a positive probability of the simultaneous survival of both alleles. Such conditions are established
in the following result.
Result 6 Let i, j > 0. If α = 0.5, min{αmR , (1−α)mR } > 1 and min{αmr , (1−
α)mr } > 1, then P(i,j) (ZRn → ∞, Zrn → ∞) > 0.
12
40000
600
30000
500
10000
20000
couples
400
300
couples
200
0
100
0
0
200
400
600
800
1000
0
200
400
generations
600
800
1000
generations
t
p
Fig. 3. Paths of the FMR (left) and FMr (right) genotypes in a process where both
have survived until generation 1000.
i
r
c
Intuitively, this result states that, for the simultaneous survival of both genotypes, the average number of male and of female descendants of each of them
must be greater than one. This happens even in the case of very unbalanced
sizes of the two genotypes. Actually, when α > 0.5 (resp. α < 0.5) and the
number of couples of each genotype grows, then the number of females (resp.
males) produced by all the mating units of each genotype is almost surely
greater than the number of males (resp. females) produced by these mating
units. Hence, the number of mating units of each genotype is determined by
the number of males (resp. females) given by all the mating units of their
genotype. Therefore, as we noted in the previous remark, each genotype must
develop by itself in order for there to be a positive probability of simultaneous
survival. Notice that this probability has not to be equal to one, i.e., there
could be some paths where both genotypes do not survive simultaneously due
to random fluctuations, even being all the thresholds parameters greater than
one.
s
u
n
a
d
e
m
t
p
e
c
c
A
When, however, α = 0.5, the probability for the number of females to exceed
that of males and the probability of there being more males than females are
the same. In order to form a conjecture as to the possible behaviour of the
process in this situation, where αmR = (1 − α)mR and αmr = (1 − α)mr , we
took both quantities to be greater than 1, performing a Monte Carlo simulation
of ten batches of 10 000 processes until generation 200, with reproduction
laws following Poisson distributions with means mR = 2.10 and mr = 2.15.
In all the simulated processes, we took (ZR0 , Zr0 ) = (50, 50). Table 2 lists
the results for the number of processes in each batch where both genotypes
have survived by a given generation. One observes how the number of such
processes decreases slowly to become stable over the course of succeeding
generations, with similar records for the ten batches. Also, Figure 4 shows
the proportion of processes among the 100000 simulated ones in which both
genotypes have survived by a given generation. Actually, this plot provides
13
batch
1
2
3
4
5
6
7
8
9
10
generation 20
9798
9809
9825
9819
9796
9803
9806
9796
9798
9809
generation 40
8112
8164
8071
8121
8084
8083
8128
8056
8063
8138
generation 60
6508
6452
6402
6516
6463
6450
6484
6408
6425
6482
generation 80
5566
5545
5553
5586
5526
5589
5634
5527
5547
5607
generation 100
5136
5150
5177
5151
5108
5183
5221
5084
5102
5203
generation 120
4953
4952
5009
4981
4937
5003
5036
4884
4931
5007
generation 140
4879
4866
4903
4895
4860
4919
4949
4808
4851
4932
generation 160
4841
4832
4873
4863
4832
4882
4907
4776
4816
4894
generation 180
4825
4818
4863
4846
4817
4869
4887
4763
i
r
c
t
p
4794
4888
generation 200 4820 4815 4856 4837 4813 4860 4881 4757 4790 4883
Table 2
Records of the number of processes in each batch where both genotypes have survived over the course of succeeding generations.
s
u
0.9
1.0
n
a
0.7
t
p
0.5
0.6
proportion
0.8
d
e
0.4
e
c
c
A
0
50
100
m
150
200
generations
Fig. 4. Proportion over the course of succeeding generations of simulated processes
in which both genotypes have survived.
an estimate of the probability that both genotypes survive until generation n,
with n = 1, 2, . . . . This probability is close to 0.475 for n large enough. Figure 5
shows the logarithm of the number of couples of each genotype from the path of
one process where both genotypes survive until generation 200. The observed
linear behaviour leads one to deduce a geometric growth over the course of
succeeding generations of the numbers of FMR and FMr mating units. We
could therefore conjecture that, in the situations with α = 0.5, αmR > 1, and
αmr > 1, not covered by Result 6, P(i,j) (ZRn → ∞, Zrn → ∞) > 0.
14
14
10
log(couples)
6
8
10
12
9
8
7
6
log(couples)
5
4
4
0
50
100
150
200
0
50
generations
100
150
200
generations
t
p
Fig. 5. Logarithm of the number of couples of the FMR (left) and FMr (right)
genotypes from the path of one process where both genotypes have survived until
generation 200, with α = 0.5, αmR > 1, and αmr > 1.
4
i
r
c
s
u
Concluding remarks
n
a
Two biological questions have been the motivation of this communication:
Yq microdeletions leading to relative male infertility, and reconstruction of
paternal lineages. Both questions were addressed as the transmission of a certain Y-linked gene or mark, with two alleles or versions, controlling characters
which are not expressed in the individual’s phenotype. These alleles could
represent the presence or the absence of the character in individuals, such as
having or not having Yq deletions or certain DNA polymorphisms.
d
e
t
p
m
We approached both situations by constructing a two-type bisexual branching
process which models the generation-by-generation evolution of the number of
carriers of such a Y-linked gene or mark in a two-sex monogamic population. In
this framework, only males carry the gene and individuals (females or males)
can mate with no more than one individual of the opposite sex in order to
produce new individuals which will be females with a given probability (sexratio), the same for all generations. Moreover, we considered that all males
have the same phenotype, and therefore a female chooses her mate blindly
without caring about which genotype he has. As we pointed out in Section 2,
monogamy is the predominant mating option in some birds and mammals, and
in particular in most human populations, and is therefore well suited to the
specific problems which motivated the present work (i.e., Yq deletions and the
history of paternal lineages). But true monogamy, as considered in this paper,
is difficult to find in nature. To relax this assumption, other mating options
could have been considered, such as polygamous mating or population-sizedependent number of couples. It seems likely that these other types of mating
would lead to different results, as indeed is the case in the monogamous mating
model with preference of females for males carrying one of the alleles of the
e
c
c
A
15
gene, studied in González et al. (2006), in relation to the simultaneous survival
of the two genotypes.
Roughly speaking, we have shown that the extinction or survival of each genotype depends on the magnitudes of the average numbers of females and males
produced by a mating unit of that genotype. Specifically, in the context of
Yq deletions, this means that if the fertility rate of the couples formed by
males with microdeletions is below a certain level, and there is no mutation,
the males with this character will disappear from the population in the long
term. On the contrary, if this fertility rate is above that level, there is a positive probability that the Yq deletions will remain in the population, even
though their prevalence may be very low. This behaviour does not depend
on the fertility rate of the couples formed by males without microdeletions.
Therefore, although there is possible competition between the two types of
males, this competition is not definitive for the presence of Yq deletions in the
long term. Similar conclusions can be applied to the extinction or survival of
paternal lineages. Indeed, this was the original problem which motivated work
on branching processes (see Galton (1873)), although the first answers to this
problem, based on the standard Galton-Watson branching model, did not take
into account such biological parameters as mating option or sex ratio.
t
p
i
r
c
s
u
n
a
Moreover, our model has shown itself to have the advantage over the WrightFisher model in allowing fluctuations of the population size. Also, as pointed
out above, it could be more useful than the Galton-Watson branching process
for studies such as that of Neves and Moreira (2006) on Y-chromosomal Adam,
because it considers a two-sex population and carefully models the mating
process.
d
e
t
p
m
For a sex-ratio of 0.5, some conclusions were established on the basis of simulations, but not analytically. Most populations in fact do not have a sex-ratio
exactly equal to 0.5, but slightly deviate from this value, and hence the analytical results of this paper apply. For a further development of this model, we
could assume that the sex-ratio is not constant, but slightly biased according
to the sex ratio in the population, with a tendency to move to compensate
any deviation from the balance between the two sexes. In order to restore this
balance, the probability for an offspring to be female in a generation could be
taken, for example, as one minus the ratio between the number of females and
the total number of individuals in the previous generation. This probability
will be greater than 0.5 when there is an excess of males and less than 0.5
when there is an excess of females. As a result, we would obtain a populationsize-dependent reproduction law. This feature has already been included in
branching model studies in both the asexual version (see, for example, Klebaner (1989)) and the bisexual version (see, for example, Xing and Wang
(2005)). Following the lines of those studies, it would therefore be feasible to
obtain analytical results under the conditions of such a scenario.
e
c
c
A
16
Other questions for future mathematical research were also illustrated via simulation, such as the extinction time, for instance. In a simulated example, we
showed that the time to extinction of a genotype can be extremely long when
the other genotype grows indefinitely. The study of the probability distributions associated with these extinction times could be very relevant from a
practical standpoint, for example in studying the problem of Y-chromosomal
Adam. The rate of growth of a genotype given its non-extinction could also
be an interesting problem to investigate. Another question for future research
is to determine the probabilities of fixation of each genotype or of the simultaneous survival of both. While difficult, the analytical development of these
questions may be feasible given the versatility of branching models in population dynamics modeling, and the many probabilistic tools available in the
branching processes literature.
5
t
p
i
r
c
Proofs
s
u
First, we provide a formal definition of the Y-linked bisexual branching process. To this end, we consider two independent sequences {(F Rn,l , MRn,l ) :
l = 1, 2, . . . ; n = 0, 1, . . .} and {(F rn,l , Mrn,l ) : l = 1, 2, . . . ; n = 0, 1, . . .} of independent, identically distributed, non-negative, and integer-valued bivariate
random vectors. Intuitively, (F Rn,l , MRn,l ) (resp. (F rn,l , Mrn,l )) represents
the number of females and males generated by the lth FMR (resp. FMr) mating unit in generation n.
n
a
d
e
m
t
p
With respect to the distribution of these reproduction vectors, we assume
Daley’s scheme (see Daley (1968)) with probability α (0 < α < 1) for a
descendant to be female. That is, the number of descendants of a FMR (resp.
FMr) couple, F R0,1 + MR0,1 (resp. F r0,1 + Mr0,1 ), is distributed according to
r
the reproduction law {pR
k }k≥0 (resp. {pk }k≥0 ) with mean mR (resp. mr ) and
probability generating function fR (·) (resp. fr (·)). Then, since the probability
of giving birth to a female is α, the conditional distribution of F R0,1 (resp.
F r0,1 ) given that F R0,1 + MR0,1 = k (resp. F r0,1 + Mr0,1 = k) is binomial
of size k and probability α, and therefore E[F R0,1 ] = αmR and E[MR0,1 ] =
(1 − α)mR (resp. E[F r0,1 ] = αmr and E[Mr0,1 ] = (1 − α)mr ).
e
c
c
A
Given the total number of FMR and FMr mating units at generation n, ZRn
and Zrn , respectively, the offspring generated by each genotype in generation
(n + 1) is specified by the formulas
(F Rn+1 , MRn+1 ) =
ZR
n
(F Rn,l , MRn,l ), (F rn+1 , Mrn+1 ) =
l=1
Zr
n
(F rn,l , Mrn,l ),
l=1
where the empty sum is assumed to be null. Intuitively, (F Rn+1 , MRn+1 ) (resp.
17
(F rn+1 , Mrn+1 )) represents the total number of females and males, respectively, given by all the FMR (FMr) mating units in generation n. Moreover,
Fn+1 = F Rn+1 + F rn+1
denotes the total number of females in generation (n + 1).
Given (Fn+1 , MRn+1 , Mrn+1 ), the total number of FMR and FMr couples in
generation n + 1 is given by the formulas
ZRn+1 = MRn+1 and Zrn+1 = Mrn+1 , if MRn+1 + Mrn+1 ≤ Fn+1
and
t
p
i
r
c
ZRn+1 = Xn+1 and Zrn+1 = Fn+1 − Xn+1 , if MRn+1 + Mrn+1 > Fn+1 ,
s
u
where Xn+1 follows a hypergeometric distribution of parameters Fn+1 , MRn+1 +
Mrn+1 , and MRn+1 (see Hush and Scovel (2005) for details about the hypergeometric distribution).
n
a
Finally, we define the σ-algebras Fn = σ((ZRk , Zrk ) : k = 0, . . . , n) and Gn =
σ((ZRk , Zrk ), (F Rk+1, MRk+1 ), (F rk+1, Mrk+1 ), k = 0, . . . , n), n = 0, 1, . . . .
d
e
Proof of Result 1
m
t
p
The proof of this result follows the same ideas as that in Theorem 4.1 of
González et al. (2006).
e
c
c
A
Proof of Result 4
We need the following lemma:
Lemma 2 Let i, j > 0. If there exists a constant A > 0 such that
or
E[ZRn+1 |(ZRn , Zrn )] ≤ ZRn a.s. on {ZRn ≥ A, Zrn ≥ A}
E[Zrn+1 |(ZRn , Zrn )] ≤ Zrn a.s. on {ZRn ≥ A, Zrn ≥ A},
then P(i,j) (ZRn → ∞, Zrn → ∞) = 0.
Proof of the Lemma
18
(1)
Given A > 0 satisfying (1), it is enough to prove that for every N > 0
P(i,j)
inf ZRn ≥ A ∩
n≥N
inf Zrn ≥ A ∩ {ZRn → ∞, Zrn → ∞} = 0.
n≥N
(2)
Fixed N > 0, define the stopping time, T (A), as T (A) = ∞ if inf n≥N ZRn ≥ A
and inf n≥N Zrn ≥ A, and T (A) = min{n ≥ N : ZRn < A or Zrn < A}
otherwise.
Define also the sequence of random variables {Yn }n≥0 as follows:
⎧
⎨ZR
N +n
Yn = ⎩
ZRT (A)
t
p
if N + n ≤ T (A)
if N + n > T (A).
i
r
c
To obtain (2), we show that {Yn }n≥0 is a non-negative supermartingale with
respect to {FN +n }n≥0 . In fact, Yn is FN +n -measurable for any n ≥ 0.
s
u
If n ≥ 0 and ZRN +k , ZrN +k ≥ A, k = 0, . . . , n, then T (A) ≥ N + n + 1, and
from (1) we obtain that
n
a
E[Yn+1 |FN +n ] = E[ZRN +n+1 |FN +n ] ≤ ZRN +n = Yn a.s.
on {ZRN +k , ZrN +k ≥ A : k = 0, . . . , n}.
d
e
m
If n ≥ 1 and for some k ∈ {1, . . . , n} it is satisfied that ZRN , . . . , ZRN +k−1 ≥
A and ZRN +k < A, or ZrN , . . . , ZrN +k−1 ≥ A and ZrN +k < A, then T (A) ≤
N + k < N + n + 1 and also
t
p
E[Yn+1 |FN +n ] = E[ZRT (A) |FN +n ] = Yn a.s.
e
c
on {ZRN , . . . , ZRN +k−1 ≥ A, ZRN +k < A} ∪ {ZrN , . . . , ZrN +k−1 ≥ A, ZrN +k <
A}. Finally, if ZRN < A or ZrN < A then T (A) = N < N + n + 1 for all
n ≥ 0, and we get that
c
A
E[Yn+1 |FN +n ] = E[ZRN |FN +n ] = Yn a.s. on {ZRN < A} ∪ {ZrN < A}.
In short, since An = {ZRN +k , ZrN +k ≥ A, k = 0, . . . , n} ∈ FN +n , one deduces
that
E[Yn+1 |FN +n ] =E[Yn+1 IAn |FN +n ] + E[Yn+1 IAcn |FN +n ]
=E[Yn+1 |FN +n ]IAn + E[Yn+1 |FN +n ]IAcn
≤ZRN +n IAn + ZRT (A) IAcn = Yn a.s.
Applying the Martingale Convergence Theorem, we obtain the almost sure
convergence of the sequence {Yn }n≥0 to a non-negative and finite limit, Y∞ ,
19
where
Y∞ =
⎧
⎨ lim
ZRn
n→∞
⎩ZR
if inf n≥N ZRn ≥ A and inf n≥N Zrn ≥ A
otherwise.
T (A)
Therefore we deduce (2) and the proof of the Lemma is complete.
Proof of the Result
(i) Assume (1 − α)mR ≤ 1. The proof is analogous for (1 − α)mr ≤ 1 because
of the symmetry of the model with respect to the two genotypes. Using the
definition of the process,
E[ZRn |Gn−1 ] =
⎧
⎨MR
n
⎩MRn
Fn
M Rn +M rn
t
p
i
r
c
if Fn ≥ MRn + Mrn
a.s.,
if Fn < MRn + Mrn
s
u
so that E[ZRn |Gn−1 ] ≤ MRn a.s. and, since Fn ⊆ Gn , n = 0, 1, . . . ,
n
a
E[ZRn |Fn−1] =E[E[ZRn |Gn−1 ] | Fn−1] ≤ E[MRn |Fn−1 ]
=(1 − α)mR ZRn−1 ≤ ZRn−1 a.s.
m
Applying the previous Lemma, we obtain that P (ZRn → ∞, Zrn → ∞) = 0.
d
e
(ii) Assume αmR < 1. The proof is analogous for αmr < 1. Assume also
α < 0.5. Otherwise the result is deduced from (i).
t
p
According to the previous Lemma we have to show (1) for some constant A.
But, using the definition of the process,
e
c
MRn Fn
I{F <M Rn +M rn }
MRn + Mrn n
MRn Fn
≤MRn I{F Rn ≥M Rn } + MRn I{F rn ≥M rn } +
I{F <M Rn +M rn } a.s.
MRn + Mrn n
(3)
E[ZRn |Gn−1 ] = MRn I{Fn ≥M Rn +M rn } +
c
A
Let us bound properly the conditional expectation given Fn−1 of each of these
summands.
For the first summand of (3), since we are assuming Daley’s scheme, we can
apply the bounds for the tails of a binomial distribution given by Okamoto’s
inequality (see Johnson et al. (1993)), and we have, for all n and k > 0,
P (F Rn ≥ MRn |F Rn + MRn = k) =P (F Rn ≥ k/2|F Rn + MRn = k)
2
≤e−k(0.5−α) = ak1 ,
20
2
with a1 = e−(0.5−α) < 1. Then, for all i , j > 0
P (F Rn ≥ MRn |ZRn−1 = i , Zrn−1 = j )
=E[P (F Rn ≥ MRn |F Rn + MRn )|ZRn−1 = i , Zrn−1 = j ]
≤E aF1 Rn +M Rn |ZRn−1 = i , Zrn−1 = j = fR (a1 )i .
(4)
Therefore, applying (4) and the Cauchy-Schwartz inequality,
E[MRn I{F Rn ≥M Rn } |Fn−1] ≤E[MRn2 |Fn−1 ]1/2 P (F Rn ≥ MRn |Fn−1 )1/2
≤ K1 ZRn−1 fR (a1 )ZRn−1 /2
a.s.
(5)
for some positive constant K1 .
i
r
c
For the second summand of (3), we can proceed analogously and obtain
s
u
E[MRn I{F rn ≥M rn } |Fn−1] ≤ K2 ZRn−1 fr (a1 )Zrn−1 /2
for some positive constant K2 .
t
p
a.s.
(6)
n
a
To bound the third summand of (3), given ε > 0, define γ1 = α(mR − ε),
γ2 = α(mR + ε), γ3 = (1 − α)(mR − ε), γ4 = (1 − α)(mR + ε), γ5 = α(mr − ε),
γ6 = α(mr + ε), γ7 = (1 − α)(mr − ε), γ8 = (1 − α)(mr + ε), and Bε =
1 + 2ε/(min{mR , mr } − ε). We take ε such that γ1 , γ5 > 0 and 0 < γ2 Bε < 1.
For each n ≥ 1, define also
d
e
m
t
p
AF R,n = {|F Rn − ZRn−1 αmR | ≤ ZRn−1 αε},
AM R,n = {|MRn − ZRn−1 (1 − α)mR | ≤ ZRn−1 (1 − α)ε},
AF r,n = {|F rn − Zrn−1 αmr | ≤ Zrn−1 αε},
AM r,n = {|Mrn − Zrn−1 (1 − α)mr | ≤ Zrn−1 (1 − α)ε}.
e
c
c
A
From now on, n will be dropped in the notation if there is no ambiguity.
Since the reproduction laws are assumed to have finite variances, an immediate
application of Chebyshev’s inequality gives
P (AcF R ∪ AcM R |ZRn−1 = i , Zrn−1 = j ) ≤
C1
i
(7)
P (AcF r ∪ AcM r |ZRn−1 = i , Zrn−1 = j ) ≤
C2
j
(8)
and
for some positive constants C1 and C2 .
21
Now, if we denote D = AF R ∩ AM R ∩ AF r ∩ AM r ,
MRn Fn
I{F <M Rn +M rn } |Fn−1
MRn + Mrn n
MRn Fn
≤E
I{Fn <M Rn +M rn } ID |Fn−1
MRn + Mrn
MRn Fn
+E
I{Fn <M Rn +M rn } IAcF R ∪AcM R |Fn−1
MRn + Mrn
MRn Fn
+E
I{Fn <M Rn +M rn } IAcF r ∪AcM r |Fn−1
MRn + Mrn
E
a.s.
(9)
Applying (7) and the Cauchy-Schwartz inequality,
t
p
MRn Fn
I{Fn <M Rn +M rn } IAcF R ∪AcM R |Fn−1 ≤ E MRn IAcF R ∪AcM R |Fn−1
MRn + Mrn
≤ E[MRn2 |Fn−1 ]1/2 P (AcF R ∪ AcM R |Fn−1)1/2
E
1/2
≤(K3 ZRn−1 )(C1 ZRn−1 )−1/2 = K3 ZRn−1
i
r
c
a.s.
s
u
for some positive constants K3 and K3 .
n
a
Analogously,
MRn Fn
−1/2
E
I{Fn <M Rn +M rn } IAcF r ∪AcM r |Fn−1 ≤ K4 ZRn−1 Zrn−1
MRn + Mrn
d
e
for some positive constant K4 .
t
p
Finally,
m
(10)
a.s.
(11)
MRn Fn
I{F <M Rn +M rn } ID |Fn−1
MRn + Mrn n
γ2 ZRn−1 + γ6 Zrn−1
≤ γ4 ZRn−1
γ3 ZRn−1 + γ7 Zrn−1
(mR + ε)ZRn−1 + (mr + ε)Zrn−1
= α(mR + ε)ZRn−1
(mR − ε)ZRn−1 + (mr − ε)Zrn−1
(mR − ε)ZRn−1 + 2εZRn−1 + (mr − ε)Zrn−1 + 2εZrn−1
= γ2 ZRn−1
(mR − ε)ZRn−1 + (mr − ε)Zrn−1
2ε
≤ γ2 ZRn−1 1 +
= γ2 Bε ZRn−1 a.s.
(12)
min{mR − ε, mr − ε}
E
e
c
c
A
Summarizing, from (5), (6), (9), (10), (11), and (12), we deduce that
E[ZRn |Fn−1 ] ≤ (K1 fR (a1 )ZRn−1 /2 + K2 fr (a1 )Zrn−1 /2
−1/2
−1/2
+ K3 ZRn−1 + K4 Zrn−1 + γ2 Bε )ZRn−1
22
a.s.
Since a1 < 1 and γ2 Bε < 1, we can take A > 0 such that, for ZRn−1 > A
and Zrn−1 > A, the term in parentheses is less than 1, so that (1) holds and
therefore the result is proved.
Proof of Result 6
For each η1 , η2 > 1, let An = {ZRn+1 > η1 ZRn , Zrn+1 > η2 Zrn } for all n ≥ 0
P(i,j) (ZRn → ∞, Zrn → ∞) ≥ P(i,j)
= lim P(i,j)
n→∞
n
l=0
∞
Al = lim P(i,j) (A0 )
n→∞
{ZRn+1 > η1 ZRn , Zrn+1 > η2 Zrn }
n=0
n
l=1
P(i,j) Al |
l−1
t
p
Ak .
(13)
i
r
c
k=0
Moreover, using the Markov property of {(ZRn , Zrn )}n≥0 , for all n > 0
P(i,j) An |
≥
=
n−1
k=0
inf
i >η1n i, j >η2n j
inf
i >η1n i, j >η2n j
⎛
Ak = P(i,j) ⎝An |
n
a
i ,j >0
P(i,j) An |{(ZRn , Zrn ) = (i , j )} ∩
P(i ,j ) (A0 ) .
s
u
{(ZRn , Zrn ) = (i , j )} ∩
d
e
m
n−1
n−1
k=0
⎞
Ak ⎠
Ak
k=0
(14)
t
p
Let us bound conveniently P(i ,j ) (A0 ) for all i , j > 0 or equivalently P(i ,j ) (Ac0 ).
First we assume that α > 0.5, and both (1 − α)mR and (1 − α)mr are greater
than 1. Take ε > 0, η1 = (1 − α)mR − ε, and η2 = (1 − α)mr − ε such that
η1 , η2 > 1. Then
e
c
c
A
Ac0 = {ZR1 ≤ η1 ZR0 } ∪ {Zr1 ≤ η2 Zr0 }
⊆ {ZR1 ≤ η1 ZR0 , MR1 > η1 ZR0 , F R1 > MR1 , F r1 > Mr1 }
∪ {MR1 ≤ η1 ZR0 } ∪ {F R1 ≤ MR1 }
∪ {Zr1 ≤ η2 Zr0 , Mr1 > η2 Zr0 , F R1 > MR1 , F r1 > Mr1 }
∪ {Mr1 ≤ η2 Zr0 } ∪ {F r1 ≤ Mr1 }.
(15)
Since ZR1 = MR1 and Zr1 = Mr1 if MR1 + Mr1 < F1 , then we have that
P(i ,j ) (ZR1 ≤ η1 ZR0 , MR1 > η1 ZR0 , F R1 > MR1 , F r1 > Mr1 ) = 0
(16)
and
P(i ,j ) (Zr1 ≤ η2 Zr0 , Mr1 > η2 Zr0 , F R1 > MR1 , F r1 > Mr1 ) = 0.
23
(17)
Moreover, since the reproduction laws are assumed to have finite variances,
by Chebyshev’s inequality, it follows that
⎛
P(i ,j ) (MR1 ≤ η1 ZR0 ) = P ⎝
⎞
i
(MRk0 − (1 − α)mR ) ≤ −εi ⎠ ≤
k=1
and analogously
P(i ,j ) (Mr1 ≤ η2 Zr0 ) ≤
C1
, (18)
i
C2
,
j
(19)
for some constants C1 , C2 > 0.
t
p
In the same way as (4), we obtain, for some constant a1 < 1, that
i
r
c
P(i ,j ) (F R1 ≤ MR1 ) ≤ fR (a1 )i and P(i ,j ) (F r1 ≤ Mr1 ) ≤ fr (a1 )j .
From (15), (16), (17), (18), (19), and (20), we have
P(i ,j ) (A0 ) = 1 − P(i ,j ) (Ac0 ) ≥ 1 −
(20)
s
u
C1 C2
− − fR (a1 )i − fr (a1 )j ,
i
j
n
a
and therefore, since η1 , η2 > 1 and fR (a1 ), fr (a1 ) < 1, from (13) and (14) it
follows that
d
e
P(i,j) (ZRn → ∞, Zrn → ∞)
n
≥ P(i,j) (A0 ) lim
n→∞
t
p
inf
l
P(i ,j ) (A0 )
l
l=1 i >η1 i, j >η2 j
n
C1
1− l −
≥ P(i,j) (A0 ) lim
n→∞
η1 i
l=1
e
c
c
A
m
C2
l
l
− fR (a1 )η1 i − fr (a1 )η2 j
η2l j
> 0,
and the proof is complete for α > 0.5.
Assume now that α < 0.5, and both αmR and αmr are greater than 1. Given
ε > 0, define the constants γ1 , . . . , γ8 and the sets AF R , AM R , AF r , AM r and
D as in the proof of Result 4. Take ε small enough so that γ2 < γ3 , γ6 < γ7 ,
and
γ1
2ε
1−
min{mR + ε, mr + ε}
and choose
> 1 , γ5
1 < η1 < γ1
2ε
1−
min{mR + ε, mr + ε}
2ε
1−
min{mR + ε, mr + ε}
24
> 1,
and
2ε
.
1 < η2 < γ5 1 −
min{mR + ε, mr + ε}
Specifically, we take η1 = γ1 (1 − 3ε/min{mR + ε, mr + ε}) and η2 = γ5 (1 −
3ε/min{mR + ε, mr + ε}.
With this notation
Ac0 ⊆ ({ZR1 ≤ η1 ZR0 } ∩ D) ∪ ({Zr1 ≤ η2 Zr0 } ∩ D) ∪ D c .
(21)
Equations (7) and (8) provide a suitable bound for P(i ,j ) (D c ), so we focus our
attention on the two first terms of (21).
t
p
Since γ2 < γ3 and γ6 < γ7 , F1 < MR1 + Mr1 on D, and, by the definition of
the process, the conditional distribution of ZR1 given G0 = σ(F1 , MR1 , Mr1 )
is hypergeometric. Hence
i
r
c
P(i ,j ) ({ZR1 ≤ η1 ZR0 } ∩ D) = E(i ,j ) [P(i ,j ) (ZR1 ≤ η1 ZR0 |G0 )ID ]
MR1 F1 =E(i ,j ) P(i ,j ) ZR1 − E(i ,j ) [ZR1 |G0 ] ≤ η1 i −
G0 ID (22)
MR1 + Mr1 s
u
n
a
But on D
η1 i −
γ1 i + γ5 j MR1 F1
≤η1 i − γ3 i MR1 + Mr1
γ4 i + γ8 j (mR − ε)i + (mr − ε)j =η1 i − γ1 i
(mR + ε)i + (mr + ε)j (mR + ε)i + (mr + ε)j − 2ε(i + j )
= η1 i − γ1 i
(mR + ε)i + (mr + ε)j 2ε
≤ η1 − γ1 1 −
i
min{mR + ε, mr + ε}
ε i
γ1 .
=−
min{mR + ε, mr + ε}
d
e
m
t
p
e
c
c
A
Let us write δ = γ1 ε/ min{mR + ε, mr + ε}. Applying the previous inequality
and the bounds for the tails of a hypergeometric distribution provided in Hush
and Scovel (2005), for i large enough, we can deduce from (22)
P(i ,j ) ({ZR1 ≤ η1 ZR0 } ∩ D)
≤E(i ,j ) P(i ,j ) ZR1 − E(i ,j ) [ZR1 |G0 ] ≤ −δi G0 ID
≤E(i ,j )
δ 2 i2 − 1
δ 2 i2 − 1
exp −2
ID ≤ exp −2 MRn + 1
γ4 i + 1
≤ K1 e−B1 i
(23)
for some positive constants K1 and B1 . Notice that, without loss of generality,
we can take i, and consequently i , large enough because all the states with
both coordinates non-null are communicating.
25
Analogously, it can be deduced that
P(i ,j ) ({Zr1 ≤ η1 Zr0 } ∩ D) ≤ K2 e−B2 j
(24)
for some positive constants K2 and B2 .
Then, taking into account the decomposition in (21), from (7), (8), (23), and
(24), we deduce that
P(i ,j ) (A0 ) = 1 − P(i ,j ) (Ac0 ) ≥ 1 − K1 e−B1 i − K2 e−B2 j −
C1 C2
− ,
i
j
t
p
and therefore, since η1 , η2 > 1 and B1 , B2 > 0, from (13) and (14) it follows
that
P(i,j) (ZRn → ∞, Zrn → ∞) ≥ P(i,j) (A0 ) lim
lim
≥ P(i,j) (A0 ) n→∞
n
l=1
n→∞
−B1 η1l i
1 − K1 e
− K2 e
d
e
i
r
c
inf
l
l
l=1 i >η1 i, j >η2 j
−B2 η2l j
s
u
C1
C2
− l − l
η1 i η2 j
P(i ,j ) (A0 )
> 0,
n
a
which completes the proof for α > 0.5.
Acknowledgement
n
m
t
p
We would like to thank the anonymous referees for their constructive comments and interesting suggestions which have improved this paper.
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